Summation by parts

"Abel transformation" redirects here. For another transformation, see Abel transform.

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. The summation by parts formula is sometimes called Abel's lemma or Abel transformation.

Note also that although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any field. It will also work when one sequence is in a vector space, and the other is in the relevant field of scalars.

For two given sequences (an){\displaystyle (a_{n})\,} and (bn){\displaystyle (b_{n})\,}, with n∈N{\displaystyle n\in \mathbb {N} }, one wants to study the sum of the following series:SN=∑n=0Nanbn{\displaystyle S_{N}=\sum _{n=0}^{N}a_{n}b_{n}}

If we define Bn=∑k=0nbk,{\displaystyle B_{n}=\sum _{k=0}^{n}b_{k},} then for every n>0,{\displaystyle n>0,\,}bn=Bn−Bn−1{\displaystyle b_{n}=B_{n}-B_{n-1}\,} and

The formula for an integration by parts is ∫abf(x)g′(x)dx=[f(x)g(x)]ab−∫abf′(x)g(x)dx{\displaystyle \int _{a}^{b}f(x)g'(x)\,dx=\left[f(x)g(x)\right]_{a}^{b}-\int _{a}^{b}f'(x)g(x)\,dx}
Beside the boundary conditions, we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( g′{\displaystyle g'\,} becomes g{\displaystyle g\,} ) and one which is differentiated ( f{\displaystyle f\,} becomes f′{\displaystyle f'\,} ).

The process of the Abel transformation is similar, since one of the two initial sequences is summed ( bn{\displaystyle b_{n}\,} becomes Bn{\displaystyle B_{n}\,} ) and the other one is differenced ( an{\displaystyle a_{n}\,} becomes an+1−an{\displaystyle a_{n+1}-a_{n}\,} ).

where a is the limit of an{\displaystyle a_{n}}. As ∑bn{\displaystyle \sum b_{n}} is convergent, BN{\displaystyle B_{N}} is bounded independently of N{\displaystyle N}, say by B{\displaystyle B}. As an−a{\displaystyle a_{n}-a} go to zero, so go the first two terms. The third term goes to zero by the Cauchy criterion for ∑bn{\displaystyle \sum b_{n}}. The remaining sum is bounded by